Civil Engineering Reference
In-Depth Information
Fig. 2.13
Substitution of variables and corresponding integration limits
Replacing
t
with
t
+
τ
,
and changing the integration limits accordingly, implies (as
illustrated in Fig. 2.13) that
TT
0
T
TT
−
τ
∫∫
∫∫ ∫∫
dt dt
=
dt d
τ
+
dt d
τ
(2.81)
12
1
1
00
−−
T
τ
0 0
and thus
0
T
T T
⎡
−
τ
⎤
1
−
i
ωτ
−
i
ωτ
()
()
()
S
lim
∫∫
Cov
e
dt d
∫∫
Cov
e
dt d
ω
=
⎢
τ
⋅
τ
+
τ
⋅
τ
⎥
x
x
1
x
1
2
T
π
T
⎢
⎥
→∞
⎣
⎦
−−
T
τ
00
0
T
⎡
⎤
1
τ
τ
⎛
⎞
⎛
⎞
i
i
()
−
ωτ
()
−
ωτ
lim
∫
1
Cov
e
d
∫
1
Cov
e
d
=
⎢
+
⋅
τ
⋅
τ
+
−
τ
⋅
τ
⎥
⎜
⎟
⎜
⎟
x
x
2
T
T
π
T
⎢ ⎝
⎠
⎝
⎠
⎥
→∞
⎣
⎦
T
0
−
T
⎛
τ
⎞
1
i
()
()
−
ωτ
S
lim
∫
1
Cov
e
d
(2.82)
⇒
ω
=
−
τ
⋅
τ
⎜
⎟
x
⎜
⎟
x
2
π
T
T
→∞
⎝
⎠
−
T
Provided the integral under the auto covariance function is finite, it is then seen that in
the limit of
T
→∞, the following is obtained
+∞
1
2
−
i
ωτ
()
()
S
∫
Cov
e
d
ω
=
τ
⋅
τ
(2.83)
x
x
π
−∞
